Question: Solve for $x$ : $ 3|x + 2| + 4 = 4|x + 2| + 3 $
Answer: Subtract $ {3|x + 2|} $ from both sides: $ \begin{eqnarray} 3|x + 2| + 4 &=& 4|x + 2| + 3 \\ \\ {- 3|x + 2|} && {- 3|x + 2|} \\ \\ 4 &=& 1|x + 2| + 3 \end{eqnarray} $ Subtract $3$ from both sides: $ \begin{eqnarray} 4 &=& 1|x + 2| + 3 \\ \\ {- 3} && {- 3} \\ \\ 1 &=& 1|x + 2| \end{eqnarray} $ Simplify: $ 1 = |x + 2| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -1 = x + 2 $ or $ 1 = x + 2 $ Solve for the solution where $x + 2$ is negative: $ - 1 = x + 2$ Subtract ${2}$ from both sides: $ \begin{eqnarray} - 1 &=& x + 2 \\ \\ {- 2} && {- 2} \\ \\ -1 - 2 &=& x \end{eqnarray} $ $ -3 = x $ Then calculate the solution where $x + 2$ is positive: $ 1 = x + 2 $ Subtract ${2}$ from both sides: $ \begin{eqnarray} 1 &=& x + 2 \\ \\ {- 2} && {- 2} \\ \\ 1 - 2 &=& x \end{eqnarray} $ $ -1 = x $ Thus, the correct answer is $x = -3 $ or $x = -1 $.